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CHAPTER 2. PRELIMINARIES 18
where e r is the r-th unity vector in R d . For ā(.; ., .) as defined above we get
Analogously B is defined by the relation
in 2D resp.
in 3D, with
and C by
A(u h ) r,s ≡ 0 if r ≠ s.
B = ( B 1 B 2)
B = ( B 1 B 2 B 3)
B r = b(ϕ j · e r , ψ k )) j,k ,
C = (c(ψ j , ψ k )) j,k .
In the same manner we define the mass matrix
the pressure mass matrix
and the Laplacian
M = ((ϕ j , ϕ k ) 0 ) j,k ,
M p = ((ψ j , ψ k ) 0 ) j,k ,
A D = (a D (ϕ j , ϕ k )) j,k ,
which we will need later in this thesis.
We denote the FE-isomorphisms between the discrete spaces and the spaces of coefficient
vectors by φ U : (R n ) d → U h (ũ 1h ) and φ Q : R m → Q h . The underline notation is used
to indicate their inverses, i.e.
φ U v h = v h , φ U v h = v h , (2.19)
φ Q q h
= q h , φ Q q h
= q h
. (2.20)
If it is clear from the context we omit the underlines and φ’s and identify v h ∈ U h (ũ 1h )
and the associated v h ∈ (R n ) d and analogously q h and q h
.
2.2.1 Examples of Mixed Elements
We present some popular choices of finite element pairs U h × Q h , in particular those we
will use later for the construction of algebraic multigrid methods and in the numerical
examples, all of them based on triangular resp. tetrahedral elements. Thus, we assume
that some partitioning of G into triangles resp. tetrahedra G = ⋃ i τ i is given, we denote
the diameter of an element τ i by h τi , we assume that we can identify some typical diameter
h (the discretization parameter) with
αh ≤ h τi ≤ ᾱh for all i,
where α and ᾱ are some positive constants, and we denote the set of elements by T h =
{τ 1 , τ 2 , . . .}. On each element τ i we define the space P k (τ i ) of polynomials of degree less
than or equal k.